Semiclassical Resolvent Estimates Research Articles

Improved resolvent bounds for radial potentials

We prove semiclassical resolvent estimates for the Schrödinger operator in \({\mathbb {R}}^d\), \(d\ge 3\), with real-valued radial potentials \(V\in L^\infty ({\mathbb {R}}^d)\). In particular, we show that if \(V(x)={{\mathcal {O}}}\left( \langle x\rangle ^{-\delta }\right) \) with \(\delta >2\), then the resolvent bound is of the form \(\exp \left( Ch^{-4/3}\right) \) with some constant \(C>0\). We also get resolvent bounds when \(1<\delta \le 2\). For slowly decaying \(\alpha \)—Hölder potentials, we get better resolvent bounds of the form \(\exp \left( Ch^{-4/(\alpha +3)}\right) \).

Semiclassical resolvent estimates for Hölder potentials

We first prove semiclassical resolvent estimates for the Schrodinger operator in R d , d ≥ 3, with real-valued potentials which are Holder with respect to the radial variable. Then we extend these resolvent estimates to exterior domains in R d , d ≥ 2, and real-valued potentials which are Holder with respect to the space variable. As an application, we obtain the rate of the decay of the local energy of the solutions to the wave equation with a refraction index which may be Holder, Lipschitz or just L ∞ .

Semiclassical Resolvent Estimates for $$L^\infty $$ Potentials on Riemannian Manifolds

We prove semi-classical resolvent estimates for the Schr{\"o}dinger operator with a real-valued L $\infty$ potential on non-compact, connected Riemannian manifolds which may have a compact smooth boundary. We show that the resolvent bound depends on the structure of the man-ifold at infinity. In particular, we show that for compactly supported real-valued L $\infty$ potentials and asymptoticaly Euclidean manifolds the resolvent bound is of the form exp(Ch --4/3 log(h --1)), while for asymptoticaly hyperbolic manifolds it is of the form exp(Ch --4/3), where C > 0 is some constant.

Semiclassical Estimates for Scattering on the Real Line

We prove explicit semiclassical resolvent estimates for an integrable potential on the real line. The proof is a comparatively easy case of the spherical energies method, which has been used to prove similar theorems in higher dimensions and in more complicated geometric situations. The novelty in our results lies in the weakness of the assumptions on the potential.

Semiclassical resolvent estimates for short-range L∞ potentials

We prove semi-classical resolvent estimates for real-valued potentials V ∈ L ∞ (R n), n ≥ 3, satisfying V (x) = O x −δ with δ > 3.

Semiclassical resolvent estimates for short-range L ∞ potentials. II

We prove semiclassical resolvent estimates for real-valued potentials V ∈ L ∞ ( R n ) , n ⩾ 3 , of the form V = V L + V S , where V L is a long-range potential which is C 1 with respect to the radial variable, while V S is a short-range potential sat

Semiclassical resolvent estimates for bounded potentials

We study the cut-off resolvent of semiclassical Schr{\"o}dinger operators on $\mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $\lambda^2$ in a compact interval in $\mathbb{R}_+$ and for any smooth cut-off function $\chi$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has \begin{equation*} \| \chi (-h^2\Delta + V-\lambda^2)^{-1} \chi \|_{L^2\to H^1} \leq C \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This bound shows in particular an upper bound on the imaginary parts of the resonances $\lambda$, defined as a pole of the meromorphic continuation of the resolvent $(-h^2\Delta + V-\lambda^2)^{-1}$ as an operator $L^2_{\mathrm{comp}}\to H^2_{\mathrm{loc}}$: any resonance $\lambda$ with real part in a compact interval away from $0$ has imaginary part at most \begin{equation*} \mathrm{Im} \lambda \leq - C^{-1} \,\mathrm{e}^{Ch^{-4/3}\log \frac{1}{h} }. \end{equation*} This is related to a conjecture by Landis: The principal Carleman estimate in our proof provides as well a lower bound on the decay rate of $L^2$ solutions $u$ to $-\Delta u = Vu$ with $0\not\equiv V\in L^{\infty}(\mathbb{R}^d)$. We show that there exist a constant $M>0$ such that for any such $u$, for $R>0$ sufficiently large, one has \begin{equation*} \int_{B(0,R+1)\backslash \overline{B(0,R)}}|u(x)|^2 dx \geq M^{-1}R^{-4/3} \mathrm{e}^{-M \|V\|_{\infty}^{2/3} R^{4/3}}\|u\|^2_2. \end{equation*}

The semiclassical resolvent on conformally compact manifolds with variable curvature at infinity

ABSTRACTWe construct a semiclassical parametrix for the resolvent of the Laplacian acting on functions on nontrapping conformally compact manifolds with variable sectional curvature at infinity. We apply this parametrix to analyze the Schwartz kernel of the semiclassical resolvent and Poisson operator and to show that the semiclassical scattering matrix is a semiclassical Fourier Integral Operator of appropriate class that quantizes the scattering relation. We also obtain high energy estimates for the resolvent and show existence of resonance free strips of arbitrary height away from the imaginary axis. We then use the results of Datchev and Vasy on gluing semiclassical resolvent estimates to obtain semiclassical resolvent estimates on certain conformally compact manifolds with hyperbolic trapping.

Analytic Continuation and Semiclassical Resolvent Estimates on Asymptotically Hyperbolic Spaces

In this paper we construct a parametrix for the high-energy asymptotics of the analytic continuation of the resolvent on a Riemannian manifold which is a small perturbation of the Poincaré metric on hyperbolic space. As a result, we obtain non-trapping high energy estimates for this analytic continuation.

Non-trapping estimates near normally hyperbolic trapping

In this paper we prove semiclassical resolvent estimates for operators with normally hyperbolic trapping which are lossless relative to non-trapping estimates but take place in weaker function spaces. In particular, we obtain non-trapping estimates in standard $L^2$ spaces for the resolvent sandwiched between operators which localize away from the trapped set $\Gamma$ in a rather weak sense, namely whose principal symbols vanish on $\Gamma$.

Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics

For a class of non-selfadjoint $h$--pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of the spectrum of the operator in an ${\cal O}(h)$--neighborhood of the origin. Moreover, when all the singular spaces are reduced to zero, we establish accurate semiclassical resolvent estimates of subelliptic type, which depend directly on algebraic properties of the Hamilton maps associated to the quadratic approximations of the principal symbol.

Semi-classical resolvent estimates for Schrödinger operators

Semi-classical resolvent estimates for Schrödinger operators

Non-Elliptic Quadratic Forms and Semiclassical Estimates for Non-selfadjoint Operators

We consider a class of pseudodifferential operators with a doubly characteristic point, where the quadratic part of the symbol fails to be elliptic but obeys an averaging assumption. Under suitable additional assumptions, semiclassical resolvent estimates are established, where the modulus of the spectral parameter is allowed to grow slightly more rapidly than the semiclassical parameter.

Propagation through trapped sets and semiclassical resolvent estimates

Motivé par l’étude des estimations de la résolvante dans la présence de capture, on démontre un théorème de propagation semiclassique dans un voisinage d’un sous-ensemble compact et invariant du flôt bicaractéristique, qui est isolé dans un sens convenable. Les exemples incluent un ensemble capté global et une trajectoire périodique isolée. Ceci est appliqué pour obtenir des estimations microlocales de la résolvante sans perte par rapport au cas non-captif.

Semiclassical resolvent estimates at trapped sets

We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs χ microlocally supported away from the trapping: ∥χR h (E+i0)χ∥=𝒪(h -1 ), a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, χ ˜, to be supported at the trapped set, giving ∥χR h (E+i0)χ ˜∥=𝒪(a(h)h -1 ) when the a priori bound is ∥χ ˜R h (E+i0)χ ˜∥=𝒪(a(h)h -1 ).

Semiclassical Resolvent Estimates in Chaotic Scattering

We prove resolvent estimates for semiclassical operators such as − h 2 Δ + V(x) in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by h − M in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrodinger propagation and to energy decay of solutions to wave equations.

Semiclassical Analysis for the Kramers–Fokker–Planck Equation

ABSTRACT On étudie des estimations semiclassiques sur la résolvente d'opérateurs qui ne sont ni elliptiques ni autoadjoints, que l'on utilise pour étudier le problème de Cauchy. En particulier on obtient une description précise du spectre pres de l'axe imaginaire, et des estimations de résolvente à l'intérieur du pseudo-spectre. On applique ensuite les résultats à l'opérateur de Kramers–Fokker–Planck. We study some accurate semiclassical resolvent estimates for operators that are neither selfadjoint nor elliptic, and applications to the Cauchy problem. In particular we get a precise description of the spectrum near the imaginary axis and precise resolvent estimates inside the pseudo-spectrum. We apply our results to the Kramers–Fokker–Planck operator.

Semi-classical resolvent estimates for the Schr�dinger operator on non-compact complete Riemannian manifolds

We prove uniform semi-classical estimates for the resolvent of the Schrodinger operator h 2Δ g + V (x), 0 0, where V is a real-valued non-negative potential and Δ g denotes the positive Laplace-Beltrami operator on a non-compact complete Riemannian manifold which may have a nonempty compact smooth boundary.

Semiclassical resolvent estimates for Schrödinger matrix operators with eigenvalues crossing

AbstractFor semiclassical Schrödinger 2×2–matrix operators, the symbol of which has crossing eigenvalues, we investigate the semiclassical Mourre theory to derive bounds O(h−1) (h being the semiclassical parameter) for the boundary values of the resolvent, viewed as bounded operator on weighted spaces. Under the non–trapping condition on the eigenvalues of the symbol and under a condition on its matricial structure, we obtain the desired bounds for codimension one crossings. For codimension two crossings, we show that a geometrical condition at the crossing must hold to get the existence of a global escape function, required by the usual semiclassical Mourre theory.

Semiclassical Estimates¶in Asymptotically Euclidean Scattering

In this note we obtain semiclassical resolvent estimates for non-trapping long range perturbations of the Laplacian on asymptotically Euclidean manifolds. Our proof is based on a positive commutator argument which differs from Mourre-type estimates by making the commutant also positive. The resolvent estimates, including the weighting of the Sobolev spaces in the estimates, are an immediate consequence.